The statistics of low-lying zeros of quadratic Dirichlet Lfunctions were conjectured by Katz and Sarnak to be given by the scaling limit of eigenvalues from the unitary symplectic ensemble. The n-level densities were found to be in agreement with this in a certain neighborhood of the origin in the Fourier domain by Rubinstein in his Ph.D. thesis in 1998. An attempt to extend the neighborhood was made in the Ph.D. thesis of Peng Gao (2005), who under GRH gave the density as a complicated combinatorial factor, but it remained open whether it coincides with the Random Matrix Theory factor. For n ≤ 7 this was recently confirmed by Levinson and Miller. We resolve this problem for all n, not by directly doing the combinatorics, but by passing to a function field analogue, of L-functions associated to hyper-elliptic curves of given genus g over a field of q elements. We show that the answer in this case coincides with Gao's combinatorial factor up to a controlled error. We then take the limit of large finite field size q → ∞ and use the Katz-Sarnak equidistribution theorem, which identifies the monodromy of the Frobenius conjugacy classes for the hyperelliptic ensemble with the group USp(2g). Further taking the limit of large genus g → ∞ allows us to identify Gao's combinatorial factor with the RMT answer.

For a projective curve⊂ defined over we study the statistics of the -structure of a section of by a random hyperplane defined over in the → ∞ limit. We obtain a very general equidistribution result for this problem. We deduce many old and new results about decomposition statistics over finite fields in this limit. Our main tool will be the calculation of the monodromy of transversal hyperplane sections of a projective curve.

We study the distribution of the zeroes of the L-functions of curves in the Artin-Schreier family. We consider the number of zeroes in short intervals and obtain partial results which agree with a random unitary matrix model.

We prove an analogue of the classical Bateman-Horn conjecture on prime values of polynomials for the ring of polynomials over a large finite field. Namely, given non-associate, irreducible, separable and monic (in the variable x) polynomialswhere N i = n deg x F i is the generic degree of F i (t, f ) for deg f = n and µ i is the number of factors into which F i splits over F q . Our proof relies on the classification of finite simple groups. We will also prove the same result for non-associate, irreducible and separable (over F q (t)) polynomials F 1 , . . . , F m not necessarily monic in x under the assumptions that n is greater than the number of geometric points of multiplicity greater than two on the (possibly reducible) affine plane curve C defined by the equation where N i is the generic degree of F i (t, f ) for deg f = n.

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